%I #30 Mar 12 2015 23:28:12
%S 1,4,13,5,35,20,86,65,194,175,14,415,430,56,844,970,182,1654,2075,490,
%T 3133,4220,1204,30,5773,8270,2716,120,10372,15665,5810,390,18240,
%U 28865,11816,1050,31449,51860,23156,2580,53292,91200,43862,5820,55,88873,157245,80822,12450,220,146095,266460,145208,25320,715
%N Triangle read by rows: T(n,k), n>=1, k>=1, in which column k lists the numbers of A210843 multiplied by A000330(k), and the first element of column k is in row A000217(k).
%C Conjecture: gives an identity for the sum of all divisors of all positive integers <= n. Alternating sum of row n equals A024916(n), i.e., sum_{k=1..A003056(n))} (-1)^(k-1)*T(n,k) = A024916(n).
%C Row n has length A003056(n) hence the first element of column k is in row A000217(k).
%C Column 1 is A210843.
%C Column k lists the partial sums of the k-th column of triangle A252117 which gives an identity for sigma.
%C The first element of column k is A000330(k).
%C The second element of column k is A002492(k).
%e Triangle begins:
%e 1;
%e 4;
%e 13, 5;
%e 35, 20;
%e 86, 65;
%e 194, 175, 14;
%e 415, 430, 56;
%e 844, 970, 182;
%e 1654, 2075, 490;
%e 3133, 4220, 1204, 30;
%e 5773, 8270, 2716, 120;
%e 10372, 15665, 5810, 390;
%e 18240, 28865, 11816, 1050;
%e 31449, 51860, 23156, 2580;
%e 53292, 91200, 43862, 5820, 55;
%e 88873, 157245, 80822, 12450, 220;
%e 146095, 266460, 145208, 25320, 715;
%e 236977, 444365, 255360, 49620, 1925;
%e 379746, 730475, 440286, 93990, 4730;
%e 601656, 1184885, 746088, 173190, 10670;
%e 943305, 1898730, 1244222, 311160, 22825, 91;
%e ...
%e For n = 6 the sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 1 + 3 + 4 + 7 + 6 + 12 = 33. On the other hand the 6th row of triangle is 194, 175, 14, so the alternating row sum is 194 - 175 + 14 = 33, equaling the sum of all divisors of all positive integers <= 6.
%Y Cf. A000203, A000217, A000330, A002492, A003056, A024916, A195825, A196020, A210843, A211970, A236104, A252117.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Dec 14 2014